Difference between revisions of "Matrix"

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(→‎Addition of matrices: rewrite this sentence)
(→‎Multiplication of matrices: rewrite this section)
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===Multiplication of matrices===
 
===Multiplication of matrices===
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To multiply two matrices, one uses the rule "go along the rows and down the columns". This is best illustrated by a specific example: a matrix times a vector:
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To multiply two matrices, use the rule for finding the product of two matrices:
  
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<math>\begin{bmatrix}
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<math>
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a & b \\
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(AB)_{ij}=\sum_k A_{ik}B_{kj}
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c & d
 
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\end{bmatrix} \begin{bmatrix}
 
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e \\ f
 
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\end{bmatrix}=\begin{bmatrix}
 
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ae+bf \\
 
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ce+df \end{bmatrix}
 
 
</math>
 
</math>
  
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It is important to note that matrix multiplication is not [[commutative]]: in general, <math>AB \neq BA</math> for two matrices <math>A</math> and <math>B</math>. This has important consequences in [[quantum mechanics]].
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However, not every pair of matrices can be multiplied.  In order for matrices <math>A</math> and <math>B</math> to be compatible for multiplication, the number of ''columns'' in <math>A</math> must equal the number of ''rows'' in <math>B</math>.  If <math>A</math> is an <math>m \times n</math> matrix and <math>B</math> is an <math>n \times p</math> matrix, the product matrix <math>AB</math> will have <math>m</math> rows and <math>p</math> columns.
  
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To see why matrix multiplication works the way it does, we will use [[suffix notation]]. Consider first forming the product of two matrices, <math>AB</math>, which is itself a matrix. Then form the product <math>ABx</math>. Matrix multiplication is associative, so we can consider this as either <math>(AB)x</math> or <math>A(Bx)</math>. In suffix notation,
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Matrix multiplication is [[associativity|associative]]. However, matrix multiplication is ''not'' [[commutativity|commutative]]. That is, it is possible for <math>AB \neq BA</math>.
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<math>
 
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\sum_j(AB)_{ij}x_j = \sum_k A_{ik}(Bx)_k=\sum_{j,k}A_{ik}B_{kj}x_j
 
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</math>
 
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The vector <math>x</math> is arbitrary, so we can therefore deduce the rule for finding the product of two matrices:
 
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<math>
 
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(AB)_{ij}=\sum_k A_{ik}B_{kj}
 
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</math>
 
  
 
===Matrix concepts===
 
===Matrix concepts===

Revision as of 00:49, July 1, 2012

For the 1999 film, see The Matrix.

A matrix (pl.: "matrices," Latin origin) is a complex ordering, in deliberate fashion, of numerals. In mathematics, a "matrix" is a regular grid of numbers, which may be manipulated and solved through intermediate-level algebra. Matrix algebra is usually taught in sophomore high school level mathematics.

More formally, a matrix is an example of a rank-2 tensor.

Alternately, a matrix may also be a complex ordering of a group of equivalent objects, especially where the order is imposed to gain incidental benefit from the synergy of the networked objects.

Mathematics

In mathematics, matrices can be manipulated in a variety of ways, including addition and multiplication.

Addition of matrices

To add two matrices, one would add their respective elements. For example:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} x & y & z \\ 1 & 3 & 5 \\ 0 & 2 & 0 \end{bmatrix} + \begin{bmatrix} 0 & 3 & 1 \\ 4 & 3 & {x+2} \\ 0 & 4 & v \end{bmatrix} }

would equal

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} {x+0} & {y+3} & {z+1} \\ {1+4} & {3+3} & {5+(x+2)} \\ {0+0} & {2+4} & {0+v} \end{bmatrix} = \begin{bmatrix} x & {y+3} & {z+1} \\ 5 & 6 & {x+7} \\ 0 & 6 & v \end{bmatrix} }

Multiplication of matrices

To multiply two matrices, use the rule for finding the product of two matrices:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (AB)_{ij}=\sum_k A_{ik}B_{kj} }

However, not every pair of matrices can be multiplied. In order for matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} to be compatible for multiplication, the number of columns in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} must equal the number of rows in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} . If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times n} matrix and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} is an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times p} matrix, the product matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AB} will have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} rows and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} columns.

Matrix multiplication is associative. However, matrix multiplication is not commutative. That is, it is possible for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AB \neq BA} .

Matrix concepts

Basic concepts

Advanced concepts

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